Subgroup Proximity in Banach Lie Groups

Let U be a Banach Lie group and G≤U a compact subgroup. We show that closed Lie subgroups of U contained in sufficiently small neighborhoods V⊇G are compact, and conjugate to subgroups of G by elements close to 1∈U; this generalizes a well-known result of Montgomery and Zippin’s from finite- to infinite-dimensional Lie groups. Along the way, we also prove an approximate counterpart to Jordan’s theorem on finite subgroups of general linear groups: finite subgroups of U contained in sufficiently small neighborhoods V⊇G have normal abelian subgroups of index bounded in terms of G≤U alone. Additionally, various spaces of compact subgroups of U, equipped with the Hausdorff metric attached to a complete metric on U, are shown to be analytic Banach manifolds; this is the case for both (a) compact groups of a given, fixed dimension, or (b) compact (possibly disconnected) semisimple subgroups. Finally, we also prove that the operation of taking the centralizer (or normalizer) of a compact subgroup of U is continuous (respectively upper semicontinuous) in the appropriate sense.